Wizards, Aliens, and Starships: Physics and Math in Fantasy and Science Fiction

by Adler, Charles L.

  CHAPTER NINETEEN

  DYSON SPHERES AND RINGWORLDS

  For all the richest and most powerful merchants life inevitably became rather dull and niggly, and they began to imagine that this was therefore the fault of the worlds they’d settled on. None of them was entirely satisfactory: either the climate wasn’t quite right in the later part of the afternoon, or the day was half an hour too long, or the sea was exactly the wrong shade of pink.

  And thus was created the conditions for a staggering new form of specialist industry: custom-made luxury planet building.

  —DOUGLAS ADAMS, THE HITCHHIKER’S GUIDE TO THE GALAXY

  19.1 DYSON’S SPHERE

  In June 1960 a paper titled “Search for Artificial Stellar Sources of Infrared Radiation” appeared in the journal Science [71]. Its author, Freeman Dyson, is a physicist who is currently a fellow of the Institute for Advanced Study at Princeton University (the same place where Einstein worked). We have encountered him before as one of the chief driving forces behind the Orion nuclear pulse propulsion drive. In the Science paper he presented an idea for finding alien civilizations that did not require searching for radio signals from them; instead, he suggested that astronomers look for sources of infrared radiation radiating at a temperature about right for life on a planet, but with the total power output of a star like our Sun.

  The material factors which ultimately limit the expansion of a technologically advanced species are the supply of materials and the supply of energy. At present the material resources being exploited by the human species are roughly limited to the biosphere of the Earth…. The quantities of matter and energy which might conceivably become accessible to us within the solar system are … the mass of Jupiter … [and] the total energy output of the sun [71].

  Let us consider the twin issues of energy and resource needs versus their supply.

  19.1.1 Energy and Resource Needs

  Figure 19.1 shows an estimate of world population since the year 14 CE. It is taken from a 1977 publication, supplemented by more recent data [69, table 5]:

  As one can see, world population increased dramatically between around 1800 CE and now, presumably owing to the Industrial Revolution. Not only did population increase dramatically, so too did the growth rate. From about 1800 to the present, world population has increased at an average annualized rate of about 1.3%. This doesn’t sound like a lot, but I’ll reintroduce a rule of estimation from an earlier chapter: the time it takes to double any quantity increasing at an annual rate of q% is 70/q years. This means that at current rates of expansion, the world population doubles every 54 years. Let’s assume there are 100,000 habitable planets in our galaxy. If we could somehow shunt our excess population to these other (hypothetical) worlds, it would take only about 900 years until all of them had populations equal to that of present-day Earth.

  Figure 19.1. World Population over Time.

  If we believe that faster-than-light travel is impossible, then this is clearly impossible. It would take far longer than 900 years simply to get to the other planets. Also, the energy requirements of placing that many people off-planet are probably impossible to meet. Freeman Dyson considered the problem of exponentially increasing population and energy demands in a 1961 paper. His question was whether one could figure out a solution to the population problem without interstellar travel.

  19.2 THE DYSON NET

  Dyson considered building numerous habitable space stations in orbit around the Sun at roughly the same distance as Earth is from the Sun. His idea for this solution came directly from science fiction. In interviews and published work he has credited Olaf Stapledon’s book Star Maker, in which the members of an intergalactic civilization build large shells around their stars to capture the last remnants of energy as the stars die out [225]. Dyson’s stations would occupy a large fraction of the total surface area of a sphere around the Sun of radius 1 AU, so that they would intercept most of the light coming from the Sun. It should be note that this network of stations is not a Dyson sphere; the Dyson sphere would be a solid shell built around the Sun that would intercept all the power coming from it. I’ll refer to the multiple space stations design as a “Dyson net” to distinguish it from a Dyson sphere. The Dyson sphere has two major problems with it:

  1. Newton’s shell theorem states there is no net force between a solid spherical shell such as a Dyson sphere and the sun it surrounds. This means there is no way to prevent the shell from drifting off-center and hitting the Sun. There are even worse problems involving Larry Niven’s Ringworld, which we look at in the next section.

  2. Even though the net force is zero, there is still a stress on the sphere that tends to compress any given section of it. This stress is much higher than any known material can support. Again, we will consider this in detail for the Ringworld in the next section.

  Large, individual space stations in orbit around the Sun don’t have this problem. Each of them is in orbit, meaning that centrifugal force balances out the force of gravity acting on them. The tricky part is preventing the hundreds of thousands of space stations in orbit around the Sun from colliding with each other.

  The amount of matter needed to build this shell of space stations is enormous. As Dyson stated, if we limit ourselves to material in the Solar System, most of the mass (outside the Sun) is concentrated in the giant planet Jupiter. Jupiter’s mass is larger than all of the other planets’ masses combined. It has a mass of 1.9×1027 kg, or roughly 318 times the mass of the Earth. Between Jupiter and Saturn we could probably scrape together about 400 ME total, or about 2.5×1027 kg of material. That is a lot, but if we spread it in a thin shell with radius 1 AU around the Sun, it would have a thickness of only about 9 meters if the density were the same as water. However, this is more than we need.

  19.2.1 Large Structures in Space

  In a later paper published in 1996, Dyson considered the engineering issues associated with building large space-based habitats for the Dyson net [74]. He examined three questions in the paper:

  1. Is it possible to build large structures in space?

  2. Is it possible to build light, rigid structures in space?

  3. Is it possible to take planets apart for the material needed to build these structures?

  The constraints on building large structures in space are different from those that apply to building large structures on the surface of a planet. The ultimate constraint on how large a space station can be made is tidal forces. In some sense we’ve encountered this constraint already in considering the space elevator, although that structure represented a rather extreme version of the issue.

  If we imagine a space station in orbit around a planet or star, there is only one point, the center of gravity, where centrifugal force balances out the attraction of gravity. Any part of the station “below” the center of mass, closer to the primary body, will see gravity as slightly larger than centrifugal force, and anything “above” it will find centrifugal force stronger than gravity. This means that there will be a seeming force trying to pull the station apart. This is what is known as a tidal force, The first time I heard of it was in Larry Niven’s short story “Neutron Star,” in which the hero of the story is nearly torn apart by these tidal forces when his spacecraft tries to get too close to said star [176].

  The net stress pulling apart a structure of size L in a circular orbit of radius r around a primary body of mass M is of order

  where g = G M/r2 is the acceleration of gravity owing to the primary and m is the mass of the space station. This is an approximation only; the stress will be a function of the shape of the craft and its distribution of mass. If the station has mean density ρ, then m ≈ ρL3. Working this out, we find that

  where Ymax is the maximum possible stress the structure can take and h is the parameter defined in chapter 5 for the space elevator: the maximum height a material of a given stress/weight ratio can stand in a gravitational field. One big difference here is that the gravitational field
is that due to the Sun rather than the Earth. Putting in numbers appropriate for steel, I get a maximum size of about 9×108 m, or nearly 106 km, which is what Dyson found in his paper as well.1 This is a structure roughly 150 times larger than the Earth. Another way to look at it is that the structure is larger than the distance of the Moon from the Earth. Large as that is, carbon nanotube fibers, with higher strength/weight ratios by a factor of 1,000 or more, would allow structures more than thirty times larger to be built.

  The distance from Earth to the Sun is 1.5×1011 m. If we wanted enough of these structures to intercept all the light from the Sun, the approximate number is

  19.2.2 Building Light Rigid Structures

  The question Dyson addressed next was whether it was possible to build light, rigid structures using the least amount of material. For structures of constant density, the mass of a structure is related to its size by the relation

  M ∼ L3.

  Because of the cubic dependence on size, mass increases rapidly as we build very large structures. Dyson considered what we would call today a “fractal,” or self-similar, structure. This is built of subunits of octahedrons built into larger octahedral units, which are built into even larger octahedral units, and so on. Because of how the structure is built, and because it doesn’t have to support its own weight when in orbit, one can build very large, very light structures. The details are in Dyson’s paper [70]. Dyson estimated that building a structure 106 km in size would require a mass of about 3×1014 kg, or about 5×10−11ME.

  Because of its fractal construction, the mass scales as a lower power of the size:

  M ∼ L3/2

  meaning that the overall density decreases as the structure gets larger. To build a Dyson shell one would need a total mass of about 105 times the mass of one of these huge space stations, or about 1020 kg.

  This is still a lot of mass. The total mass of the Empire State Building is about 4×108 kg, so the Dyson net represents about a trillion copies of it put into space. This is a mass much larger than that of all the buildings in all the cities in the world put together.

  19.2.3 Taking Planets Apart

  The total mass required to build a Dyson net, while less than the mass of the Earth, is still huge by any standard. Could we imagine taking the Earth or another planet apart to rebuild it into these structures?

  The first question to ask is how much energy it takes to deconstruct a planet. The gravitational potential energy of a uniform sphere is

  A planet isn’t a uniform sphere, but we’ll take this as a starting point. In the formula, Mp is the planet’s mass and Rp is its radius. Putting in values appropriate for Earth, we get U = −3.8×1031 J. In normalized units, we can write this (for any planet) as

  in units where M is the mass of the planet relative to Earth’s mass and R is the radius relative to Earth’s. At least this much energy must be supplied to break up the planet. The luminosity of the sun is 3.86×1026 J/s; the energy it would take to disassemble Earth represents the total output from the Sun for a day.

  This leads to the question of how to do it. Dyson imagined an ingenious scheme to increase the Earth’s rotational speed until the planet flew apart. He proposed making an electrical motor out of the Earth. Laying large wires parallel to lines of latitude across it and running a current through them would give the Earth a sizable magnetic field. Then, running a current from pole to pole and out to large distances from the Earth and back again, one could generate a sizable torque either to speed it up or slow it down. Dyson envisioned doing this over the course of 40,000 years, which would require an average power of about 300 times the total power from the Sun intercepted by the Earth.

  In Greg Bear’s Forge of God, hostile aliens destroy Earth much more rapidly by dropping two large, ultradense masses into the center of the Earth. One is matter, the other antimatter, both at neutronium densities. When they merged together, the resulting explosion destroyed the world [32]. Based on the calculation above and using E = Mc2, it would require a mass of about 1014 kg to do this. The purpose of this was destructive rather than constructive, but the net effect was the same: a planet in pieces.

  19.2.4 Detection of a Dyson Net

  Dysons claimed that his motivation for writing these two papers was not to suggest that humanity should actually build such a structure. To quote Dyson,

  When one discusses engineering projects on the grand scale, one can either think of what we, the human species, may do here in the future, or one can think of what extraterrestrial species, if they exist, may have already done elsewhere. To think about a grandiose future for humanity … is to pursue idle dreams…. But to think in a disciplined way about what we may be able to observe now astronomically … is a serious and legitimate form of science [74].

  He wanted to see if astronomers could detect such structures if they existed. The issue is that the power output from a star like our Sun follows a blackbody curve with a characteristic temperature of about 6,000 K. This means that the peak in the spectrum is at a wavelength of about 0.5 µm, or 5×10−7 m. However, the Dyson net’s mean temperature would be a lot lower, around 300 K. The shell would absorb the sunlight at a relatively high temperature and reradiate it at a lower temperature. An astronomer would see a blackbody curve with a peak wavelength of about 10 µm, or 10−5 m. There are other astronomical sources of radiation at this wavelength. However, unlike other sources, it would look like a star with the same luminosity as our Sun but with a planet-like temperature. In fact, one can argue that if a civilization didn’t completely surround the star, astronomers would see two blackbody curves of similar strength: one resulting from the partly blocked star itself at a high temperature, and one from the shell at a much lower temperature. This seems a reasonable and unique signature for such a structure. I say “reasonable” because it is probably impossible to block the star entirely. The Dyson net isn’t a solid shell but a large number of relatively small satellites in orbit around it. To keep the satellites from colliding, there should be some room between them. To spot a Dyson net, one might look for stars like our Sun that radiate most of their energy in the visible region of the spectrum, but also a relatively large fraction (perhaps 10%–50%) in the far infrared. I can’t think of any naturally occurring astronomical objects that have a spectrum like that, but perhaps there are some.

  19.3 NIVEN’S RINGWORLD

  There are obvious issues with building a Dyson shell: the energy it requires, the need to destroy a world to create it, and so on. In 1970 Larry Niven came up with a smaller structure than a Dyson sphere for his novel Ringworld [177]. As the name implies, the Ringworld is a ring around a star, of radius 1 AU and width to be determined. Niven decided to spin it to provide gravity (in the form of centrifugal force) for its inhabitants.

  What I’d like to do in this section is to play “Ringworld engineer”—that is, go through the process that (presumably) Larry Niven did when originally designing the thing. It’s a fun exercise, and the concept reflects the best of what hard science fiction has to offer; of course, our job here is much easier than Niven’s was: we have only to reconstruct the idea, not come up with the original notion. One other thing: I am deliberately not referring back to the book for the parameters but will estimate them based on what I know of physics and astronomy. At the end, we’ll compare them to the ones found in the book.

  19.3.1 Ringworld Mass

  First, let’s estimate its mass. It is considerably larger than a planet but smaller than a star. Unlike the Dyson net, it is a rigid structure. Oddly enough, its mass will be larger than what we calculated for the Dyson shell.

  It’s larger than a planet but smaller than a star. Perhaps we can use the geometric mean of the mass of Earth and the Sun for its mass. Since the Earth’s mass is roughly 10−6 of the Sun’s mass, the geometric mean of the two masses is roughly 1/1,000 the Sun’s mass, or 1,000 times the mass of the Earth: about 6×1027 kg. Another way to estimate the mass is to try to figure out where all this mass
should come from. If we assume the designers used only the resources available in one stellar system, then we should estimate the Ringworld mass as the mass of everything in the Solar System but the Sun. As I stated above, most of the mass in the Solar System apart from the Sun is in the planet Jupiter. The combined mass of Jupiter and Saturn is roughly 390 ME, which is not too far from my first estimate. Of course, astronomers have found “super-Jupiters” in other stellar systems with masses more than ten times Jupiter’s mass, so there are good reasons to think that I can be liberal by a factor of two or even more. I will use an estimate of 5×1027 kg to make the calculations ahead come out nicer.

  19.3.2 Ringworld Radius and Mean Temperature

  One parameter of importance is the radius of this ring around the Sun. Because the Ringworld is inhabited by humans and humanoid aliens such as Kzinti, the distance from its star must be about the same as the distance from Earth to the Sun. If not, the radiant flux from the star will make the world too hot or too cold. On this assumption, R ≈1 AU = 1.5×1011 m. There is one caveat: given this distance, one might naively expect the Ringworld temperature to about the same as Earth’s, but this is ignoring the fact that greenhouse warming by Earth’s atmosphere leads to a significant rise in Earth’s mean temperature (roughly 30°C). “So what?,” I hear the fan say: “Ringworld’s atmosphere, just about the same composition as Earth’s, leads to greenhouse warming of the Ringworld as well.” Yes, but Earth’s atmosphere extends completely around the planet, whereas Ringworld’s atmosphere is only on the inner side. One must apply the principle of detailed balance to calculate the temperature, using the fact that part of the heat flux escaping from the structure leaves through the back. Let’s calculate the mean temperature of the Ringworld using the principle of detailed balance.